Question: The sum of two numbers is $117$, and their difference is $21$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 117}$ ${x-y = 21}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 138 $ $ x = \dfrac{138}{2} $ ${x = 69}$ Now that you know ${x = 69}$ , plug it back into $ {x+y = 117}$ to find $y$ ${(69)}{ + y = 117}$ ${y = 48}$ You can also plug ${x = 69}$ into $ {x-y = 21}$ and get the same answer for $y$ ${(69)}{ - y = 21}$ ${y = 48}$ Therefore, the larger number is $69$, and the smaller number is $48$.